Honestly I don't know why you can't.
To enter you just need to click the Training Center door (your mouse pointer changes from 'Hand' into 'Shoes' when you hover on it)
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I had 116 diamonds, after a fight in the dark forest, I have 56 of them ...
Also, you should precise which type of game we play in each part of the world, maybe in the training center.
In every battle field you can see a symbol in shield shape at the center bottom of the screen. This tells you which style you need to use. In Training Center you learn which symbol stays for which style.
And as for click area in Town Map, yes, you're right, we'll fix it in the next release. Thanks.
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Oh yeah, found it. The fact is that the clickable zones are nexte to the labels on the other spots, and fot the training center, you have to click on the entrance door, which is far away from the label.